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Econ-577 · economics · Economics

Exchange-rate forecasting

LLM status Open unverified

Statement

Original subject group: International finance and exchange rates

Audit classification: Background; proposed extension. The classic paper reports a historical forecasting comparison, while the 2024 IMF paper presents a conditional forecasting improvement. Neither certifies the exact broad catalogue performance target as unresolved.

Definitions and precise research target

Fix a prespecified set of currency pairs $C$, forecast horizons $H$, evaluation dates $T$, and an information set available in real time. Let $s_{ct}$ be the logarithm of currency pair $c$'s spot exchange rate. A forecasting procedure $f$ maps dated observations to $\widehat s^f_{c,t+h}$, with parameter estimation using only information available by $t$. The no-drift random-walk benchmark is $\widehat s^{RW}_{c,t+h}=s_{ct}$. For a declared loss function $\ell$, define \[\Delta_{ch}(f)=E[\ell(s_{c,t+h},\widehat s^f_{c,t+h})-\ell(s_{c,t+h},s_{ct})].\] The task is to identify methods and economic conditions with negative, statistically supported loss differences on untouched observations. Specify whether the claim requires improvement for every $(c,h)\in C\times H$ or only a weighted aggregate; these are different research targets. Account for predictor revisions, model selection, overlapping forecast errors and structural breaks. Compare statistical accuracy with feasible trading gains after costs rather than treating them as identical. Derive confidence sets or predictive guarantees under stated stability assumptions and test sensitivity across regimes. A method that beats the benchmark in one sample is a resolved subcase, not a universal solution; neither the classic comparison nor the present task asserts impossibility of improved forecasting.

Additional formal definitions and scope (editorial). For squared-error evaluation, require $E[(s_{c,t+h}-\widehat s^f_{c,t+h})^2]<\infty$ for each pair and horizon. An evaluation date belongs to a fixed future distribution or sampling design rather than an unspecified expectation over a historical table. A forecasting algorithm includes training windows, tuning and predictor-release calendars, and is measurable with respect to the real-time information $\mathcal I_t$. Universal improvement means $\Delta_{ch}<0$ for every declared $(c,h)$; weighted improvement means $\sum_{c,h}w_{ch}\Delta_{ch}<0$, $w_{ch}\geq0$, $\sum w_{ch}=1$. Neither follows from a unit root test: an integrated process can have predictable stationary components, and an efficient market need not make exchange rates a no-drift random walk. Guarantees require an explicit class of stable or changing laws; unrestricted distributions can make the benchmark already optimal.

For the identification part, declare an admissible class $\Theta$ of structural models, including preferences, technologies, shock laws, measurement/sampling rules and any equilibrium selector. If $O$ denotes the complete observable history and $T(\theta)$ the target defined above, the identified set is $\mathcal I_T=\{T(\theta):\theta\in\Theta,\ \mathcal L_\theta(O)=\mathcal L_{\rm obs}(O)\}$; point identification means this set is a singleton. A $do(a)$ comparison replaces stated policy/technology equations while fixing the declared exogenous law and re-solving the remaining equations. These definitions do not assert that an identification theorem holds without further restrictions.

Variants and assumption changes

  1. Fixed pair and horizon (editorial). Test a preregistered real-time forecasting rule against the no-drift benchmark for one $(c,h)$ on a fixed untouched evaluation distribution.
  2. Cross-regime predictive guarantees (editorial). Define a class of changing laws and a weighted set of pairs/horizons; seek uniform or bounded-regret performance under those restrictions rather than unrestricted universal success.

References and source audit

  1. Author publication record verifies Meese and Rogoff, 1983, Journal of International Economics 14:3--24; publisher indexed abstract verifies DOI/PII. Locator: Journal abstract, p.3. Support: Historical forecast comparison, not impossibility. Publisher direct fetch failed; author-hosted scan opened, without extracted text. URL: https://rogoff.scholars.harvard.edu/publications/empirical-exchange-rate-models-seventies-do-they-fit-out-sample.
  2. Primary IMF WP/24/252 cover verifies Bas B. Bakker, December 2024; DOI associated with official WP. Locator: Abstract; Sections 9--10, printed pp.35--39. Support: Conditional forecasting solution. Full PDF accessible; broad exact open status not asserted. URL: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024252-print-pdf.pdf.
  1. Richard A. Meese; Kenneth Rogoff. 1983. Empirical Exchange Rate Models of the Seventies: Do They Fit Out of Sample?. Journal abstract, p.3. https://rogoff.scholars.harvard.edu/publications/empirical-exchange-rate-models-seventies-do-they-fit-out-sample. DOI: https://doi.org/10.1016/0022-1996(83)90017-X.
  2. Bas B. Bakker. 2024. Reconciling Random Walks and Predictability: A Dual-Component Model of Exchange Rate Dynamics. Abstract; Sections 9--10, printed pp.35--39. https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024252-print-pdf.pdf. DOI: https://doi.org/10.5089/9798400295034.001.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-577-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-577

Exchange-rate forecasting: predictable components and honest delayed comparisons

Target and scope

For a prespecified currency pair and horizon $h$, the squared-loss target relative to the no-drift random walk is \[ \Delta_h(f)=E[(s_{t+h}-\widehat s^f_{t+h})^2-(s_{t+h}-s_t)^2]. \] Negative loss difference is a statistical statement about a declared evaluation law. It is not equivalent to stationarity or profitable trading. Bakker's primary working paper [exchange] explicitly combines permanent and transitory components; its conditional forecasting findings do not establish universal improvement. This note derives a transparent gain formula, quantifies predictor noise, and gives a sequential comparison allowing overlapping horizons.

An integrated process can be predictably forecastable

Consider \[ s_t=R_t+z_t,\qquad R_{t+1}=R_t+\eta_{t+1},\qquad z_{t+1}=\phi z_t+\varepsilon_{t+1},\quad |\phi|<1. \] The two innovation sequences are mutually independent, centered, square integrable, and independent of the real-time past. Assume $z_t$ is stationary with variance $V=\sigma_\varepsilon^2/(1-\phi^2)$. All parameters below are supplied to isolate the forecasting question. Let the information set include $s_t$ and satisfy the stated future-innovation independence, and put $m_t=E[z_t\mid\mathcal I_t]$.
Theorem (The exact squared-loss gain): The conditional-mean forecast is \[ s_t+(\phi^h-1)m_t. \] Its expected loss improvement over the random walk is $(1-\phi^h)^2E[m_t^2]$. If $z_t$ itself is observed, its mean squared error is \[ h\sigma_\eta^2+\sigma_\varepsilon^2\sum_{j=0}^{h-1}\phi^{2j}, \] whereas random-walk mean squared error adds $(1-\phi^h)^2V$.
Proof: Iteration gives $s_{t+h}-s_t=(\phi^h-1)z_t+\sum_{j=1}^h\eta_{t+j}+\sum_{j=1}^h\phi^{h-j}\varepsilon_{t+j}$. Conditional expectation annihilates future innovations and replaces $z_t$ by $m_t$. For any square-integrable outcome $U$, conditional-mean projection gives $E(U-g)^2=E(U-E[U\mid\mathcal I_t])^2+E(g-E[U\mid\mathcal I_t])^2$ for every information-measurable forecast $g$. Apply this to the random-walk forecast to obtain the gain. When $z_t$ is observed, only the future innovation sums remain in the optimal error; independence gives the displayed variance. The random-walk error also contains $(\phi^h-1)z_t$, independent of those sums, adding the stated term. ∎
If the stationary component is absent or conditionally unpredictable, the gain is zero. In the pure random-walk submodel any forecast differing from $s_t$ has excess risk equal to the mean square of that difference. Thus a model class containing that submodel cannot promise strict improvement everywhere without further restrictions.
Proposition (Noisy predictors require shrinkage): Suppose a supplied real-time predictor is $v_t=z_t+\xi_t$, where $\xi_t$ is centered, independent of $z_t$ and future innovations, with variance $\nu$. Consider forecasts $\widehat s_{t+h}=s_t-\lambda(1-\phi^h)v_t$. Their loss difference is \[ \Delta_h(\lambda)=(1-\phi^h)^2\{\lambda^2(V+\nu)-2\lambda V\}. \] For $V+\nu>0$, the best coefficient is $\lambda^*=V/(V+\nu)$, with improvement $(1-\phi^h)^2V^2/(V+\nu)$. If $V>0$, improvement occurs exactly for $0<\lambda<2V/(V+\nu)$.
Proof: Write the future change as $-(1-\phi^h)z_t+u_{t,h}$, with $u_{t,h}$ centered and independent of $z_t,\xi_t$. Subtract the random-walk squared error from that of the candidate. Taking expectations cancels $Eu_{t,h}^2$ and all innovation cross terms, leaving $(1-\phi^h)^2E[(z_t-\lambda v_t)^2-z_t^2]$. The covariance $E[z_tv_t]=V$ and variance $E[v_t^2]=V+\nu$ give the formula. Minimizing the quadratic and solving its strict-negativity inequality prove the remaining claims. ∎
Consequently a theoretically relevant predictor can worsen forecasting when used without sufficient shrinkage. Parameter learning and predictor revisions are not free: any operational rule must compute its coefficient and predictor using information available at the forecast date.

A delayed, overlap-aware comparison

Fix a finite pool of $M$ prespecified currency--horizon--algorithm comparisons, each with real-time forecasts. For one horizon, let $g_t=\widehat s_{t+h}-s_t$ and $U_t=s_{t+h}-s_t$. Assume the declared evaluation class satisfies $|g_t|,|U_t|\le B$ almost surely; these are bounded-law restrictions, not claims about unrestricted exchange rates. The realized loss difference is $d_t=g_t^2-2g_tU_t$, so $|d_t|\le3B^2$. Put $\mu_t=E[d_t\mid\mathcal I_t]$ and $\overline\mu=T^{-1}\sum_t\mu_t$. Labels arrive after $h$ periods and are available before the next forecast at that same date.
Theorem (An honest interval for average conditional loss): Let $k=\min(h,T)$. Simultaneously for the $M$ predeclared comparisons, with probability at least $1-\alpha$, \[ |\overline d-\overline\mu| \le6B^2\sqrt{\frac{k\log(2kM/\alpha)}{2T}}. \] The event holds for adaptive real-time forecasting rules and overlapping forecast errors under the stated conditional boundedness, without independence across consecutive dates.
Proof: Partition dates into $k$ residue subsequences. Along each, $d_t-\mu_t$ is a martingale difference: $g_t$ is measurable at prediction and the preceding label in that subsequence is then observed. Its conditional range has length at most $6B^2$. The conditional Hoeffding exponential bound, iterated along a subsequence of length $n_r$, gives $|\sum_{t\in r}(d_t-\mu_t)|\le6B^2\sqrt{n_r\log(2kM/\alpha)/2}$ except with probability $\alpha/(kM)$. Union bound across subsequences and comparisons, then use $\sum_r\sqrt{n_r}\le\sqrt{kT}$ and divide by $T$. If comparisons have different $k$, assigning failure $\alpha/M$ separately to each still gives the stated simultaneous conclusion. ∎
A negative upper endpoint certifies negative average conditional loss on this evaluation path. This is explicitly not a confidence interval for a fixed unconditional stationary mean without an additional stability or sampling argument. Under iid nonoverlapping evaluation blocks, an ordinary bounded-mean interval instead estimates that fixed expectation. Unbounded squared errors require moment or tail assumptions and a different concentration argument. Searching over unregistered algorithms invalidates the finite-pool guarantee unless the enlarged search is also covered.

Accuracy and trading are different targets

For a separately specified funded currency or forward payoff $R_{t+1}$ per unit position, let a predictable position $a_t\in[-1,1]$ incur proportional transaction cost $c|a_t|$. Conditional expected net payoff is $a_tE[R_{t+1}\mid\mathcal I_t]-c|a_t|$. Maximizing this piecewise linear expression gives $\max\{|E[R_{t+1}\mid\mathcal I_t]|-c,0\}$. Thus a predictable signal can have a positive squared-forecast gain while all feasible positions in this trading class have nonpositive expected net gain if conditional expected payoffs never exceed costs. Log-rate forecasts, currency excess returns, funding rates and utility risk adjustments require their own mapping; none is silently equated here.

Remaining gap

The note proves improvement under a supplied predictable-component model and gives an honest restricted evaluation procedure. It does not establish that any particular currency has those stable primitives, learn them optimally across breaks, or demonstrate empirical improvement on untouched observations. Uniform claims across pairs, aggregate gains and implementable trading benefits therefore remain separate unresolved targets.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves forecasting gains in a supplied predictable-component model, optimal shrinkage for noisy predictors, and an overlap-aware guarantee for prespecified real-time comparisons. It has not established stable currency-specific primitives or untouched-sample improvement across the declared pairs and horizons, and break adaptation and implementable trading gains remain open.

References

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